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A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

%I A318773 #17 Aug 08 2025 23:29:04
%S A318773 1,3,9,27,81,1,243,6,729,27,2187,108,6561,405,1,19683,1458,9,59049,
%T A318773 5103,54,177147,17496,270,531441,59049,1215,1,1594323,196830,5103,12,
%U A318773 4782969,649539,20412,90,14348907,2125764,78732,540,43046721,6908733,295245,2835,1,129140163,22320522,1082565,13608,15
%N A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
%C A318773 The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
%C A318773 The coefficients in the expansion of 1/(1-3*x-x^4) are given by the sequence generated by the row sums.
%C A318773 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.035744112294..., when n approaches infinity.
%D A318773 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
%H A318773 G. C. Greubel, <a href="/A318773/b318773.txt">Rows n = 0..120 of the irregular triangle, flattened</a>
%H A318773 Zagros Lalo, <a href="/A318773/a318773_1.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n</a>
%H A318773 Zagros Lalo, <a href="/A318773/a318773.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n</a>
%F A318773 T(n,k) = 3^(n-4*k) * (n-3*k)!/(k! * (n-4*k)!) where n >= 0 and 0 <= k <= floor(n/4).
%e A318773 Triangle begins:
%e A318773           1;
%e A318773           3;
%e A318773           9;
%e A318773          27;
%e A318773          81,        1;
%e A318773         243,        6;
%e A318773         729,       27;
%e A318773        2187,      108;
%e A318773        6561,      405,       1;
%e A318773       19683,     1458,       9;
%e A318773       59049,     5103,      54;
%e A318773      177147,    17496,     270;
%e A318773      531441,    59049,    1215,     1;
%e A318773     1594323,   196830,    5103,    12;
%e A318773     4782969,   649539,   20412,    90;
%e A318773    14348907,  2125764,   78732,   540;
%e A318773    43046721,  6908733,  295245,  2835,   1;
%e A318773   129140163, 22320522, 1082565, 13608,  15;
%e A318773   387420489, 71744535, 3897234, 61236, 135;
%e A318773   ...
%t A318773 T[n_, k_]:= T[n, k] = 3^(n-4k)*(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]} ]//Flatten
%t A318773 T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3T[n-1, k] + T[n-4, k-1]]; Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]}]//Flatten
%o A318773 (Magma) [3^(n-4*k)*Binomial(n-3*k,k): k in [0..Floor(n/4)], n in [0..24]]; // _G. C. Greubel_, May 12 2021
%o A318773 (Sage) flatten([[3^(n-4*k)*binomial(n-3*k,k) for k in (0..n//4)] for n in (0..24)]) # _G. C. Greubel_, May 12 2021
%Y A318773 Row sums give A052917.
%Y A318773 Cf. A013610, A027465.
%Y A318773 Cf. A304236, A304249
%Y A318773 Cf. A317496, A317497.
%Y A318773 Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3).
%Y A318773 Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), A317497 (q=3), this sequence (q=4).
%K A318773 tabf,nonn,easy
%O A318773 0,2
%A A318773 _Zagros Lalo_, Sep 04 2018